## Eigenvalue estimates and nodal length of eigenfunctions.(English)Zbl 1055.58015

Kozma, L. (ed.) et al., Steps in differential geometry. Proceedings of the colloquium on differential geometry, Debrecen, Hungary, July 25–30, 2000. Debrecen: Univ. Debrecen, Institute of Mathematics and Informatics. 295-301 (2001).
The paper under review is a nice and clear survey (without proof) of the following result: Let $$M$$ be a 2-dimensional compact, smooth Riemannian manifold without boundary, and let $$\Phi$$ be an eigenfunction associated to the eigenvalue $$\lambda$$. Then the bound: $\text{Length}[\Phi ^{-1}(0)] > \frac{1}{11} \text{Area}(M) \sqrt{\lambda}$ holds if $$\lambda$$ is large. If the curvature of $$M$$ is everywhere non-negative, then the bound holds for all eigenvalues.
Details can be found in Ann. Global Anal. Geom. 19, No. 2, 133–151 (2001; Zbl 1010.58025).
For the entire collection see [Zbl 0966.00031].

### MSC:

 58J50 Spectral problems; spectral geometry; scattering theory on manifolds 53C20 Global Riemannian geometry, including pinching

### Keywords:

Laplace operator; nodal sets; Riemann surfaces

Zbl 1010.58025
Full Text: